Joint Athens-Atlanta Number Theory Seminar

The Joint Athens-Atlanta Number Theory Seminar meets once a semester, usually on a Tuesday, with two talks back to back, at 4:00 and at 5:15. Participants then go to dinner together.

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Fall 2024

TBD, at the University of Georgia

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Spring 2024

Tuesday April  16, at Emory (in Atwood 240)

First talk at 4:00 by Jiuya Wang (UGA)

Dominant Galois Groups for Large Degree Number Field

 

Second talk at 5:15 by Andrew Obus (CUNY)

Regular models of superelliptic curves via Mac Lane valuations

Let X --> P^1 be a Z/n-branched cover over a complete discretely valued field K, where n does not divide the residue characteristic of K.  We explicitly construct the minimal regular normal crossings model of X over the valuation ring of K.  By “explicitly”, we mean that we construct a normal model of P^1 whose normalization in K(X) is the desired regular model.  The normal model of P^1 is fully encoded as a basket of finitely many discrete valuations on the rational function field K(P^1), each of which is given using Mac Lane’s 1936 notation involving finitely many polynomials and rational numbers.  This is joint work with Padmavathi Srinivasan.

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Fall 2023

October 24, Room 006 in the Skiles building at Georgia Tech

First talk at 4:00 by Caroline Turnage-Butterbaugh (Carleton) 

Moments of Dirichlet L-functions

Second talk at 5:15 by Vesselin Dimitrov (Georgia Tech)

Sqrt and Levers

"If the formal square root of an abelian surface over Q looks like an elliptic curve, it has to be an elliptic curve." We discuss what such a proposition might mean, and prove the most straightforward version where the precise condition is simply that the L-function of the abelian surface possesses an entire holomorphic square root. The approach follows the Diophantine principle that algebraic numbers or zeros of L-functions repel each other, and is in some sense similar in spirit to the Gelfond--Linnik--Baker solution of the class number one problem.  

We discuss furthermore this latter connection: the problems that it raises under a hypothetical presence of Siegel zeros, and a proven analog over finite fields. The basic remark that underlies and motivates these researches is the well-known principle (which is a consequence of the Deuring--Heilbronn phenomenon, to be taken with suitable automorphic forms f and g): an exceptional character \chi would cause the formal \sqrt{L(s,f)L(s,f \otimes \chi)}L(s,g)L(s, g \otimes \chi) to have a holomorphic branch on an abnormally big part of the complex plane, all the while enjoying a Dirichlet series formal expansion with almost-integer coefficients. This leads to the kind of situation oftentimes amenable to arithmetic algebraization methods. The most basic (qualitative) form of our main tool is what we are calling the "integral converse theorem for GL(2)," and it is a refinement of a recent Unbounded Denominators theorem that we proved jointly with Frank Calegari and Yunqing Tang. 

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Spring 2023

Monday February 6, at UGA, Boyd 328. Refreshments at 3:30 in the departmental lounge on the 4th floor.

First talk at 4:00 by Andrew Granville (University of Montreal)

What makes a proof acceptable?

In the first year of university mathematics we are taught about the axiomatic structure of mathematics, how we build a solid structure of correct statements and their proofs, that can lead us to the stars (modulo Godel). We are taught that every correct statement can (and should) be justified back to the axioms.

Working research mathematicians build their work on libraries of published material and hope that everything quoted is justifiable back to the axioms, so that their new contribution is also. This is also how new work is judged so if the system works perfectly then this satisfies the axiomatic dream, even though most researchers have little interest in going back to the axioms themselves, indeed much of the work that would be necessary to do so would be irrelevant.

However we are all familiar with proofs that are a little shaky, perhaps missing a few details for us to be 100% sure, or even with a hole that "seems fillable". Professional mathematicians work with a notion of "robustness of proof", that even if a few details are awry one can be confident of the big structure. Thus even statements with a (formally) incorrect proof form part of the accepted literature, which seems like an odd way to proceed.

How should the successes of proof verification software affect the community's view of what constitutes a correct proof? How will it? In this talk we bring together several threads to investigate these questions, and propose some objectives for the future.

Second talk at 5:15 by Carl Pomerance (Dartmouth)

Digits  

A low-brow area in number theory is to consider problems based on the digits of a number. However, there has been some highly nontrivial and interesting work involving digits.  This talk will survey some of these problems and results.

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Fall 2022

Monday Nov 7, at Emory (MSCW301)

First talk at 4:00 by Samit Dasgupta (Duke)

Stark’s Conjectures and Hilbert’s 12th Problem  

In this talk we will discuss two central problems in algebraic number theory and their interconnections: explicit class field theory (also known as Hilbert’s 12th Problem), and the special values of L-functions. The goal of explicit class field theory is to describe the abelian extensions of a ground number field via analytic means intrinsic to the ground field. Meanwhile, there is an abundance of conjectures on the special values of L-functions at certain integer points. Of these, Stark’s Conjecture has special relevance toward explicit class field theory. This conjecture states the existence of certain canonical elements in CM abelian extensions of totally real fields. I will state a conjectural exact formula for Brumer-Stark units that has been developed over the last 15 years. I will conclude with a description of my work with Mahesh Kakde that proves these conjectures away from p=2, thereby giving an explicit class field theory for totally real fields.

Second talk at 5:15 by Alina Bucur (UCSD)

Counting D4 fields  

A guiding question in number theory, specifically in arithmetic statistics, is that of counting number fields of fixed degree whose normal closure has a given Galois group G as we let their discriminants grow to infinity. In this talk, we will discuss the history of this question and take a closer look at the story in the case that n=4, i.e. the counts of quartic fields.

 

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Spring 2020, Fall 2020, Spring 2021, Fall 2021, Spring 2022

Cancelled due to the pandemic.

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Fall 2019

Tuesday September 24, 2019, at UGA (in Boyd 304)

First talk at 4:00 by Jennifer Balakrishnan (Boston University)

A tale of three curves  

We will describe variants of the Chabauty--Coleman methodand quadratic Chabauty to determine rational points on curves. In so doing, we will highlight some recent examples where the techniques have been used: this includes a problem of Diophantus originally solved by Wetherell and the problem of the "cursed curve", the split Cartan modular curve of level 13. This is joint work with Netan Dogra, Steffen Mueller, Jan Tuitman, and Jan Vonk.  
 

Second talk at 5:15 by Dimitris Koukoulopoulos (University of Montreal)

On the Duffin-Schaeffer conjecture  

Let S be a sequence of integers. We wish to understand how well we can approximate a ``typical'' real number using reduced fractions whose denominator lies in S. To this end, we associate to each  q in S an acceptable error delta_q>0. When is it true that almost all real numbers (in the Lebesgue sense) admit an infinite number of reduced rational approximations a/q, q in S, within distance delta_q? In 1941, Duffin and Schaeffer proposed a simple criterion to decided whether this is case: they conjectured that the answer to the above question is affirmative precisely when the series sum_{q\in S} \phi(q)\delta_q diverges, where phi(q) denotes Euler's totient function. Otherwise, the set of ``approximable'' real numbers has null measure. In this talk, I will present recent joint work with James Maynard that settles the conjecture of Duffin and Schaeffer.

 

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Spring 2019

Tuesday April 23, 2019, at Georgia Tech (in Skiles 311, 3 floors above the regular seminar room)

First talk at 4:00 by Ananth Shankar (MIT)

Exceptional splitting of abelian surfaces over global function fields.

Let A denote a non-constant ordinary abelian surface over a global function field (of characteristic p > 2) with good reduction everywhere. Suppose that $A$ does not have real multiplication by any real quadratic field with discriminant a multiple of $p$. Then we prove that there are infinitely many places modulo which $A$ is isogenous to the product of two elliptic curves. If time permits, I will also talk about applications of our results to the p-adic monodromy of such abelian surfaces. This is joint work with Davesh Maulik and Yunqing Tang.

Second talk at 5:15 by Jordan Ellenberg (University of Wisconsin)

What is the tropical Ceresa class and what should it be?

This is a highly preliminary talk about joint work with Daniel Corey and Wanlin Li.  The Ceresa cycle is an algebraic cycle canonically attached to a curve C, which appears in an intriguing variety of contexts; its height can sometimes be interpreted as a special value, the vanishing of its cycle class is related to the Galois action on the nilpotent fundamental group, it vanishes on hyperelliptic curves, etc.  In practice it is not easy to compute, and we do not in fact know an explicit non-hyperelliptic curve whose Ceresa class vanishes.  We will discuss a definition of the Ceresa class for a tropical curve, explain how to compute it in certain simple cases, and describe progress towards understanding whether it is possible for the Ceresa class of a non-hyperelliptic tropical curve to vanish.  (The answer is:  "sort of”.)  The tropical Ceresa class sits at the interface of algebraic geometry, graph theory (because a tropical curve is more or less a metric graph), and topology: for we can also frame the tropical Ceresa class as an entity governed by the mapping class group, and in particular by the question of when a product of commuting Dehn twists can commute with a hyperelliptic involution in the quotient of the mapping class group by the Johnson kernel.

 

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Fall 2018

Tuesday October 23, 2018, at Emory, in W301 in the Math and Science Center (the usual building)

First talk at 4:00 by Bianca Viray (University of Washington)

On the level of modular curves that give rise to sporadic j-invariants

Merel's Uniform Boundedness Theorem states that the torsion on an elliptic curve over a number field k can be bounded by a constant that depends only on the degree [k:Q].  This theorem can be rephrased as saying that for any positive integer d, the infinite tower of modular curves {X_1(n)}_{n} has only finitely many closed points of degree at most d.  Work of Frey and Abramovich from around the same time combine to give an independent proof of a  weaker result, that for any positive integer d, there are only finitely many positive integers n such that X_1(n) has infinitely many degree d points.  In this talk, we study complementary part of Merel's theorem, that is, the points x on X_1(n) where there are only finitely many points of degree at most deg(x).  We show that these so-called sporadic points map down to sporadic points on X_1(d), where d is a bounded divisor of n.  This is joint work with A. Bourdon, Ö. Ejder, Y. Liu, and F. Odumodu.

Second talk at 5:15 by Larry Rolen (Vanderbilt)

Locally harmonic Maass forms and central L-values

In this talk, we will discuss a relatively new modular-type object known as a locally harmonic Maass form. We will discuss recent joint work with Ehlen, Guerzhoy, and Kane with applications to the theory of $L$-functions. In particular, we find finite formulas for certain twisted central $L$-values of a family of elliptic curves in terms of finite sums over canonical binary quadratic forms. Applications to the congruent number problem will be given.

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Spring 2018

Tuesday February 20, 2018, at UGA, in Boyd Room 304

First talk at 4:00 by David Harbater (University of Pennsylvania)

Local-global principles for zero-cycles over semi-global fields

Classical local-global principles are given over global fields.  This talk will discuss such principles over semi-global fields, which are function fields of curves defined over a complete discretely valued field.  Paralleling a result that Y. Liang proved over number fields, we prove a local-global principle for zero-cycles on varieties over semi-global fields.  This builds on earlier work about local-global principles for rational points.  (Joint work with J.-L. Colliot-Thélène, J. Hartmann, D. Krashen, R. Parimala, J. Suresh.)

Second talk at 5:15 by Jacob Tsimerman (U. Toronto)

Cohen-Lenstra heuristics in the Presence of Roots of Unity

The class group is a natural abelian group one can associated to a number field, and it is natural to ask how it varies in families. Cohen and Lenstra famously proposed a model for families of quadratic fields based on random matrices of large rank, and this was later generalized by Cohen-Martinet to general number fields. However, their model was observed by Malle to have issues when the base field contains roots of unity. We explain that in this setting there are naturally defined additional invariants on the class group, and based on this we propose a refined model in the number field setting rooted in random matrix theory. Our conjecture is based on keeping track not only of the underlying group structure, but also certain natural pairings one can define in the presence of roots of unity. Specifically, if the base field contains roots of unity, we keep track of the class group G together with a naturally defined homomorphism G*[n] --> G from the n-torsion of the Pontryagin dual of G to G. Using methods of Ellenberg-Venkatesh-Westerland, we can prove some of our conjecture in the function field setting.

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Fall 2017

Monday October 30, 2017, at Georgia Tech.

First talk at 4:00 by Bjorn Poonen (MIT)

Gonality and the strong uniform boundedness conjecture for periodic points

The function field case of the strong uniform boundedness conjecture for torsion points on elliptic curves reduces to showing that classical modular curves have gonality tending to infinity. We prove an analogue for periodic points of polynomials under iteration by studying the geometry of analogous curves called dynatomic curves. This is joint work with John R. Doyle.

Second talk at 5:15 by Spencer Bloch (U. Chicago)

Periods, motivic Gamma functions, and Hodge structures

Golyshev and Zagier found an interesting new source of periods associated to (eventually inhomogeneous) solutions generated by the Frobenius method for Picard Fuchs equations in the neighborhood of singular points with maximum unipotent monodromy. I will explain how this works, and how one can associate "motivic Gamma functions" and generalized Beilinson style variations of mixed Hodge structure to these solutions. This is joint work with M. Vlasenko.

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Spring 2017

Tuesday April 18, 2017, at Emory.

First talk at 4:00 by Rachel Pries (Colorado State)

Galois action on homology of Fermat curves

We prove a result about the Galois module structure of the Fermat curve using commutative algebra, number theory, and algebraic topology.  Specifically, we extend work of Anderson about the action of the absolute Galois group of a cyclotomic field on a relative homology group of the Fermat curve.  By finding explicit formulae for this action, we determine the maps between several Galois cohomology groups which arise in connection with obstructions for rational points on the generalized Jacobian.  Heisenberg extensions play a key role in the result. This is joint work with R. Davis, V. Stojanoska, and K. Wickelgren.

Second talk at 5:15 by Gopal Prasad (U. Michigan)

Weakly commensurable Zariski-dense subgroups of semi-simple groups and isospectral locally symmetric spaces 

I will discuss the notion of weak commensurability of  Zariski-dense subgroups of semi-simple groups. This notion was introduced in my joint work with Andrei Rapinchuk (Publ. Math. IHES 109(2009), 113-184), where we determined when two Zariski-dense S-arithmetic subgroups of absolutely almost simple algebraic groups over a field of characteristic zero can be weakly commensurable. These results enabled us to prove that in many situations isospectral locally symmetric spaces of simple real algebraic groups are necessarily commensurable. This settled the famous question "Can one hear the shape of a drum?" of Mark Kac for these spaces. The arguments use algebraic and transcendental number theory.  

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Fall 2016

Tuesday October 18, 2016, at UGA.

First talk at 4:00 by Florian Pop (University of Pennsylvania)

Local section conjectures and Artin-Schreier theorems

After a short introduction to (Grothendieck's) section conjecture (SC), I will explain how the classical Artin Schreier Thm and its p-adic analog imply the birational local SC; further, I will mention briefly local-global aspects of the birational SC. Finally, I will give an effective "minimalistic" p-adic Artin-Schreier Thm, which is similar in flavor to the classical Artin-Schreier Thm.

Second talk at 5:15 by Ben Bakker (UGA)

Recovering elliptic curves from their p-torsion

Given an elliptic curve E over the rationals Q, its p-torsion E[p] gives a 2-dimensional representation of the Galois group G_Q over F_p.  The Frey-Mazur conjecture asserts that for p>17, this representation is essentially a complete invariant:  E is determined up to isogeny by E[p].  In joint work with J. Tsimerman, we prove the analog of the Frey-Mazur conjecture over characteristic 0 function fields.  The proof uses the hyperbolic repulsion of special subvarieties in a modular surface to show that families of elliptic curves with many isogenous fibers have large volume.  We will also explain how these ideas relate to other uniformity conjectures about the size of monodromy representations

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Spring 2016

Thursday April 14, 2016, at Georgia Tech.

First talk at 4:00 by Melanie Matchett-Wood (University of Wisconsin)

Nonabelian Cohen-Lenstra Heuristics and Function Field Theorems

The Cohen-Lenstra Heuristics conjecturally give the distribution of class groups of imaginary quadratic fields. Since, by class field theory, the class group is the Galois group of the maximal unramified abelian extension, we can consider the Galois group of the maximal unramified extension as a non-abelian generalization of the class group. We will explain non-abelian analogs of the Cohen-Lenstra heuristics due to Boston, Bush, and Hajir and joint work with Boston proving cases of the non-abelian conjectures in the function field analog.

Second talk at 5:15 by Zhiwei Yun (Stanford University)

Intersection numbers and higher derivatives of L-functions for function fields

In joint work with Wei Zhang, we prove a higher derivative analogue of the Waldspurger formula and the Gross-Zagier formula in the function field setting under the assumption that the relevant objects are everywhere unramified. Our formula relates the self-intersection number of certain cycles on the moduli of Shtukas for GL(2) to higher derivatives of automorphic L-functions for GL(2).

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Fall 2015

Tuesday November 17, 2015, at Emory.

First talk at 4:00 by John Duncan (Emory)

K3 Surfaces, Mock Modular Forms and the Conway Group

In their famous “Monstrous Moonshine” paper of 1979, Conway—Norton also described an association of modular functions to the automorphism group of the Leech lattice (a.k.a. Conway’s group). In analogy with the monstrous case, there is a distinguished vertex operator superalgebra that realizes these functions explicitly. More recently, it has come to light that this Conway moonshine module may be used to compute equivariant enumerative invariants of K3 surfaces. Conjecturally, all such invariants can be computed in this way. The construction attaches explicitly computable mock modular forms to automorphisms of K3 surfaces.  
One expects the Brauer-Manin obstruction to control rational points on 1-parameter families of conics and quadrics over a number field when the base curve has genus 0. Results in this direction have recently been obtained as a consequence of progress inanalytic number theory. On the other hand, it is easy to construct a family of 2-dimensional quadrics over a curve with just one rational point over Q, which is a counterexample to the Hasse principle not detected by the \'etale Brauer-Manin obstruction. Conic bundles with similar properties exist over real quadratic fields, though most certainly not over Q.

Second talk at 5:15 by Alexei Skorobogatov (Imperial College)

Local-to-global principle for rational points on conic and quadric bundles over curves

One expects the Brauer-Manin obstruction to control rational points on 1-parameter families of conics and quadrics over a number field when the base curve has genus 0. Results in this direction have recently been obtained as a consequence of progress inanalytic number theory. On the other hand, it is easy to construct a family of 2-dimensional quadrics over a curve with just one rational point over Q, which is a counterexample to the Hasse principle not detected by the \'etale Brauer-Manin obstruction. Conic bundles with similar properties exist over real quadratic fields, though most certainly not over Q.

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Spring 2015

Thursday April 9, 2015, at UGA, Room 304 in Boyd Graduate Studies Building.

First talk at 4:00 by Dick Gross (Harvard)

Pencils of quadrics

Quadric hypersurfaces, defined by homogeneous equations of degree 2, are the simplest projective varieties other than linear subspaces. In this talk I will review the theory of quadratic forms over a general field, and discuss the smooth intersection of two quadric hypersurfaces in projective space. The Fano scheme of maximal linear subspaces contained in this intersection is either finite or is a principal homogeneous space for the Jacobian of a hyperelliptic curve.  This gives an important tool for the arithmetic study of these curves.

Second talk at 5:15 by Ted Chinburg (University of Pennsylvania)

When is an error term not really an error term?

The classical case of Iwasawa theory has to do with how quickly the p-parts of the ideal class groups of number fields grow in certain towers of number fields.  I will discuss the connection of these growth rates to Chern classes, and how the "error terms" in various formulas have to do with higher Chern classes. I will then describe a result linking second Chern classes in Iwasawa theory over imaginary quadratic fields to pairs of p-adic L-functions.  This is joint work with F. Bleher, R. Greenberg, M. Kakde, G. Pappas, R Sharifi and M. Taylor.

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Fall 2014

Tuesday, November 4, 2014, at Georgia Tech.

First talk at 4:00 by Arul Shankar (Harvard University)

Geometry-of-numbers methods over number fields

We discuss the necessary modifications required to apply Bhargava's geometry-of-numbers methods to representations over number fields. As an application, we derive upper bounds on the average rank of elliptic curves over any number field. This is joint work with Manjul Bhargava and Xiaoheng Jerry Wang.

Second talk at 5:15 by Wei Zhang (Columbia University)

Kolyvagin's conjecture on Heegner points

We recall a conjecture of Kolyvagin on Heegner points for elliptic curves of arbitrary analytic rank, and present some recent results on this conjecture for elliptic curves satisfying some technical conditions.

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Spring 2014

Tuesday, February 25, 2014, at Emory.

First talk at 4:00 by Paul Pollack (UGA)

Solved and unsolved problems in elementary number theory

This will be a survey of certain easy-to-understand problems in elementary number theory about which "not enough" is known. We will start with a discussion of the infinitude of primes, then discuss the ancient concept of perfect numbers (and related notions), and then branch off into other realms as the spirit of Paul Erdös leads us.

Second talk at 5:15 by James Maynard (Université de Montréal)

Bounded gaps between primes

It is believed that there should be infinitely many pairs of primes which differ by 2; this is the famous twin prime conjecture. More generally, it is believed that for every positive integer $m$ there should be infinitely many sets of $m$ primes, with each set contained in an interval of size roughly $m\log{m}$. We will introduce a refinement of the `GPY sieve method' for studying these problems. This refinement will allow us to show (amongst other things) that $\liminf_n(p_{n+m}-p_n)<\infty$ for any integer $m$, and so there are infinitely many bounded length intervals containing $m$ primes.

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Fall 2013

Tuesday, November 5, 2013, at UGA, Room 303 in Boyd Graduate Studies Building.

First talk at 4:00 by Joe Rabinoff (Georgia Tech)

Lifting covers of metrized complexes to covers of curves

Let K be a complete and algebraically closed non-Archimedean field and let X be a smooth K-curve.  Its Berkovich analytification X^an deformation retracts onto a metric graph Gamma, called a skeleton of X^an.  The collection of different skeleta of X are in natural bijective correspondence with the collection of semistable models of X over the valuation ring R of K.  We prove that, given a finite morphism of curves f: Y -> X, there exists a skeleton Gamma_X of X^an whose inverse image is a skeleton Gamma_Y of Y^an.  This can be seen as a "skeletal" simultaneous semistable reduction theorem, which can in fact be used to give new, more general proofs of foundational results of Liu, Liu-Lorenzini, and Coleman on simultaneous semistable reductions.  We then consider the following problem: given X, a skeleton Gamma_X, and a finite harmonic morphism of metric graphs Gamma' -> Gamma_X, can we find a curve Y and a finite morphism f: Y -> X such that f^{-1}(Gamma_X) is a skeleton of Y^an and is isomorphic to Gamma'?  In general the answer is no: one must enrich the skeleton with the structure of a metrized complex of curves.  In this context the answer is yes, and moreover the map Gamma' -> Gamma_X can be used to calculate the finitely many isomorphism classes of Y -> X, as well as their automorphisms.  We give an application to component groups of Jacobians, answering a question of Ribet.

Second talk at 5:15 by Kirsten Wickelgren (Georgia Tech)

Splitting varieties for triple Massey products in Galois cohomology

The Brauer-Severi variety a x^2 + b y^2 = z^2 has a rational point if and only if the cup product of cohomology classes associated to a and b vanish. The cup product is the order-2 Massey product. Higher Massey products give further structure to Galois cohomology, and more generally, they measure information carried in a differential graded algebra which can be lost on passing to the associated cohomology ring. For example, the cohomology of the Borromean rings is isomorphic to that of three unlinked circles, but non-trivial Massey products of elements of H^1 detect the more complicated structure of the Borromean rings. Analogues of this example exist in Galois cohomology due to work of Morishita, Vogel, and others. This talk will first introduce Massey products and some relationships with non-abelian cohomology. We will then show that b x^2 = (y_1^2 - ay_2^2 + c y_3^2 - ac y_4^2)^2 - c(2 y_1 y_3 - 2 a y_2 y_4)^2 is a splitting variety for the triple Massey product <a,b,c>, and that this variety satisfies the Hasse principle. The method could produce splitting varieties for higher order Massey products. It follows that all triple Massey products over global fields vanish when they are defined. More generally, one can show this vanishing over any field of characteristic different from 2; Jan Minac and Nguyen Duy Tan, and independently Suresh Venapally, found an explicit rational point on X(a,b,c). Minac and Tan have other nice results in this direction.  This is joint work with Michael Hopkins.

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Spring 2013

Tuesday, April 16, 2013, at Georgia Tech

First talk at 4:00 by Dick Gross (Harvard)

The arithmetic of hyperelliptic curves  

Hyperelliptic curves over Q have equations of the form y^2 = F(x), where F(x) is a polynomial with rational coefficients which has simple roots over the complex numbers. When the degree of F(x) is at least 5, the genus of the hyperelliptic curve is at least 2 and Faltings has proved that there are only finitely many rational solutions. In this talk, I will describe methods which Manjul Bhargava and I have developed to quantify this result, on average.

Second talk at 5:15 by Jordan Ellenberg (Wisconsin)

Arithmetic statistics over function fields

What is the probability that a random integer is squarefree? Prime? How many number fields of degree d are there with discriminant at most X? What does the class group of a random quadratic field look like? These questions, and many more like them, are part of the very active subject of arithmetic statistics. Many aspects of the subject are well-understood, but many more remain the subject of conjectures, by Cohen-Lenstra, Malle, Bhargava, Batyrev-Manin, and others. In this talk, I explain what arithmetic statistics looks like when we start from the  field Fq(x) of rational functions over a finite field instead of the field Q of rational numbers. The analogy between function  fields and number  fields has been a rich source of insights throughout the modern history of number theory. In this setting, the analogy reveals a surprising relationship between conjectures in number theory and conjectures in topology about stable cohomology of moduli spaces, especially spaces related to Artin's braid group. I will discuss some recent work in this area, in which new theorems about the topology of moduli spaces lead to proofs of arithmetic conjectures over function fields, and to new, topologically motivated questions about counting arithmetic objects.

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Fall 2012

Thursday, October 25, 2012, at Emory.

First talk at 4:00 by Karl Rubin (UCI)

Ranks of elliptic curves

I will discuss some recent conjectures and results on the distribution of Mordell-Weil ranks and Selmer ranks of elliptic curves. After some general background, I will specialize to families of quadratic twists, and describe some recent results in detail.

Second talk at 5:15 by Jayce Getz (Duke)

An approach to nonsolvable base change for GL(2)

Motivated by Langlands' beyond endoscopy idea, the speaker will present a conjectural trace identity that is essentially equivalent to base change and descent of automorphic representations of GL(2) along a nonsolvable extension of fields.

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Spring 2012

Tuesday, April 10, 2012, at UGA, Room 323 in Boyd Graduate Studies Building

First talk at 4:00 by Max Lieblich (Univ. of Washington)

Finiteness of K3 surfaces and the Tate conjecture

Fix a finite field k. It is well known that there are only finitely many smooth projective curves of a given genus over k. It turns out that there are also only a finite number of abelian varieties of a given dimension over k. What about other classes of varieties? I will review the history of these results and describe joint work with Maulik and Snowden that links the finiteness of K3 surfaces over k to the Tate conjecture for K3 surfaces over k. The key is a link between certain lattices in the l-adic cohomology of K3 surfaces and derived categories of sheaves on certain algebraic stacks. I will not assume you know anything about any of this.

Second talk at 5:15 by Frank Calegari (Northwestern Univ.)

Even Galois Representations

What Galois representations "come" from algebraic geometry? The Fontaine-Mazur conjecture gives a very precise conjectural answer to this question. A simplified version of this conjecture in the case of two dimensional representations says that "all nice representations come from modular forms". Yet, by construction, all representations coming from modular forms are "odd", that is, complex conjugation acts by a 2x2 matrix of determinant -1. What happened to all the even Galois representations?

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Fall 2011   

Wednesday, November 2, 2011, at Georgia Tech in Skiles room 005 (ground floor).

First talk at 4:00 by Jared Weinstein (IAS)

Maximal varieties over finite fields

This is joint work with Mitya Boyarchenko.  We construct a special hypersurface X over a finite field, which has the property of "maximality", meaning that it has the maximum number of rational points relative to its topology.  Our variety is derived from a certain unipotent algebraic group, in an analogous manner as Deligne-Lusztig varieties are derived from reductive algebraic groups. As a consequence, the cohomology of X can be shown to realize a piece of the local Langlands correspondence for certain wild Weil parameters of low conductor.

Second talk at 5:15 by David Brown (Emory)

Random Dieudonne modules and the Cohen-Lenstra conjectures.

Knowledge of the distribution of class groups is elusive -- it is not even known if there are infinitely many number fields with trivial class group. Cohen and Lenstra noticed a strange pattern --experimentally, the group $\mathbb{Z}/(9)$ appears more often than $\mathbb{Z{/(3) \times \mathbb{Z}/(3)$ as the 3-part of the classgroup of a real quadratic field $\Q(\sqrt{d})$ - and refined this observation into concise conjectures on the manner in which class groups behave randomly. Their heuristic says roughly that $p$-parts of class groups behave like random finite abelian $p$-groups, rather than like random numbers; in particular, when counting one should weight by the size of the automorphism group, which explains why $\mathbb{Z}/(3) \times \mathbb{Z}/(3)$ appears much less often than $\mathbb{Z}/(9)$ (in addition to many other experimental observations).

While proof of the Cohen-Lenstra conjectures remains inaccessible, the function field analogue -- e.g., distribution of class groups of quadratic extensions of \mathbb{F}_p(t) -- is more tractable. Friedman and Washington modeled the $\ell$-power part  (with $\ell \neq p) of such class groups as random matrices and derived heuristics which agree with experiment. Later, Achter refined these heuristics, and many cases have been proved (Achter, Ellenberg and Venkatesh).

When $\ell = p$, the $\ell$-power torsion of abelian varieties, and thus the random matrix model, goes haywire. I will explain the correct linear algebraic model -- Dieudone\'e modules. Our main result is an analogue of the Cohen-Lenstra/Friedman-Washington heuristics – a theorem about the distributions of class numbers of Dieudone\'e modules (and other invariants particular to $\ell = p$). Finally, I'll present experimental evidence which mostly agrees with our heuristics and explain the connection with rational points on varieties.

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Spring 2011

Tuesday, February 1, 2011, at Emory

First talk at 4:00 by K. Soundararajan (Stanford)

Moments of zeta and L-functions

An important theme in number theory is to understand the values taken by the Riemann zeta-function and related L-functions. While much progress has been made, many of the basic questions remain unanswered. I will discuss what is known about this question, explaining in particular the work of Selberg, random matrix theory and the moment conjectures of Keating and Snaith, and recent progress towards estimating the moments of zeta and L-functions.

Second talk at 5:15 by Matthew Baker (Georgia Institute of Technology)

Complex dynamics and adelic potential theory

I will discuss the following theorem: for any fixed complex numbers a and b, the set of complex numbers c for which both a and b both have finite orbit under iteration of the map z -->z^2 + c is infinite if and only if a^2 = b^2. I will explain the motivation for this result and give an outline of the proof. The main arithmetic ingredient in the proof is an adelic equidistribution theorem for preperiodic points over product formula fields, with non-archimedean Berkovich spaces playing an essential role. This is joint work with Laura DeMarco, relying on earlier joint work with Robert Rumely.

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Fall 2010

Tuesday, September 21, 2010, at UGA

First talk at 4:00 by Ken Ono (Emory)

Mock modular periods and L-values

Recent works have shed light on the enigmatic mock theta functions of Ramanujan. These strange power series are now known to be pieces of special "harmonic" Maass forms. The speaker will discuss recent joint work in the subject with regard to special values of L-functions. This will include the study of values and derivatives of elliptic curve L-functions, as well as general critical values of modular L-functions. In addition, the speaker will derive new Eichler-Shimura isomorphisms, and will derive new relations among the "even" periods of modular L-functions. This is joint work with Jan Bruinier, Kathrin Bringmann, Zach Kent, and Pavel Guerzhoy.

Second talk at 5:15 by Armand Brumer (Fordham)

Abelian Surfaces and Siegel Paramodular Forms

This expository talk will survey recent progress on modularity of abelian surfaces. After a brief review of the history, I'll describe work of Cris Poor and David Yuen on the modular side and Ken Kramer and me on the arithmetic side.

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Spring 2010

Tuesday, April 13, 2010

First talk at 4:00 by Venapally Suresh (Emory)

Degree three cohomology of function fields of surfaces

Let k be a global field or a local field. Class field theory says that every central division algebra over k is cyclic. Let l be a prime not equal to the characteristic of k. If k contains a primitive l-th root of unity, then this leads to the fact that every element in H^2(k, µ_l ) is a symbol. A natural question is a higher dimensional analogue of this result: Let F be a function field in one variable over k which contains a primitive l-th root of unity. Is every element in H^3(F, µ_l ) a symbol? In this talk we answer this question in affirmative for k a p-adic field or a global field of positive characteristic. The main tool is a certain local global principle for elements of H^3(F, µ_l ) in terms of symbols in H^2(F µ_l ). We also show that this local-global principle is equivalent to the vanishing of certain unramified cohomology groups of 3-folds over finite fields.

Second talk at 5:15 by Antoine Chambert-Loir (IAS and University of Rennes)

Some applications of potential theory to number theoretical problems on analytic curves

Slides available at http://perso.univ-rennes1.fr/antoine.chambert-loir/publications/pdf/atlanta2010.pdf

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Fall 2009

Tuesday, October 20, 2009

First talk at 4:00 by Doug Ulmer (GA Tech).

Constructing elliptic curves of high rank over function fields

There are now several constructions of elliptic curves of high rank over function fields, most involving high-tech things like L- functions, cohomology, and the Tate or BSD conjectures. I'll review some of this and then give a very down-to-earth, low-tech construction of elliptic curves of high ranks over the rational function field Fp(t).

Second talk at 5:15 by Jonathan Hanke (UGA).

Using Mass formulas to Enumerate Definite Quadratic Forms of Class Number One

This talk will describe some recent results using exact mass formulas to determine all definite quadratic forms of small class number in n>=3 variables, particularly those of class number one. The mass of a quadratic form connects the class number (i.e. number of classes in the genus) of a quadratic form with the volume of its adelic stabilizer, and is explicitly computable in terms of special values of zeta functions. Comparing this with known results about the sizes of automorphism groups, one can make precise statements about the growth of the class number, and in principle determine those quadratic forms of small class number. We will describe some known results about masses and class numbers (over number fields), then present some new computational work over the rational numbers, and perhaps over some totally real number fields.